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Transfer entropy is a measure of the magnitude and the direction ofinformation flow between jointly distributed stochastic processes. In recentyears, its permutation analogues are considered in the literature to estimatethe transfer entropy by counting the number of occurrences of orderings ofvalues, not the values themselves. It has been suggested that the method ofpermutation is easy to implement, computationally low cost and robust to noisewhen applying to real world time series data. In this paper, we initiate atheoretical treatment of the corresponding rates. In particular, we considerthe transfer entropy rate and its permutation analogue, the symbolic transferentropy rate, and show that they are equal for any bivariate finite-alphabetstationary ergodic Markov process. This result is an illustration of theduality method introduced in [T. Haruna and K. Nakajima, Physica D 240, 1370(2011)]. We also discuss the relationship among the transfer entropy rate, thetime-delayed mutual information rate and their permutation analogues.

Symbolic transfer entropy rate is equal to transfer entropy rate for bivariate finite-alphabet stationary ergodic Markov processes